Why can no one in sci.math understand my simple point?
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From: K_h on 18 Jun 2010 22:00 "Peter Webb" <webbfamily(a)DIESPAMDIEoptusnet.com.au> wrote in message news:4c1b2def$0$14086$afc38c87(a)news.optusnet.com.au... > > "Tim Little" <tim(a)little-possums.net> wrote in message > news:slrni1m68o.jrj.tim(a)soprano.little-possums.net... >> On 2010-06-18, Peter Webb >> <webbfamily(a)DIESPAMDIEoptusnet.com.au> wrote: >>> Because Cantor's proof requires an explicit listing. >>> This is a very >>> central concept. >> >> Cantor's proof works on any list, explicit or not. >> > > Really? > > How do you apply Cantor's proof to a list constructed as > follows: > > "Define a list L such that the n'th entry on the list > consists of all 1's if the n'th digit of Omega is 1, > otherwise it is > all 0's." > > (Your example). > >> The rest of your misconception snipped. >> >> >> - Tim > > Perhaps if you could point out to me why you believe > Cantor's proof that not all Reals can be listed (as it > appears you do) but you don't believe my proof that not > all computable Reals can be listed. They appear identical > to me. All computable reals can be listed, but there is no finite algorithm for doing so. An "infinite algorithm" could list every computable real. An anti-diagonal, then, could be generated from this list but the algorithm creating the anti-diagonal is implicitly relying on the "infinite algorithm" underlying the list. In that sense the anti-diagonal is not computable. The set of all reals are a different story. Even with an "infinite algorithm" generating a list of reals, there is no way such a list could contain every real. For a proof, do a google search on Cantor's theorem. _
From: Tim Little on 18 Jun 2010 22:12 On 2010-06-19, Peter Webb <webbfamily(a)DIESPAMDIEoptusnet.com.au> wrote: > Bijectable with N and "listable" are not the same. To be "listable" > the set must be countable and recursively enumerable. There is no such requirement for recursive enumerability in Cantor's work. The concept was not even introduced into mathematics until some time after his death. Your insistence that Cantor required lists to be recursively enumerable is bizarre. - Tim
From: Tim Little on 18 Jun 2010 22:14 On 2010-06-19, Peter Webb <webbfamily(a)DIESPAMDIEoptusnet.com.au> wrote: > That, BTW, is my own interpretation of what is happening. It is an incorrect interpretation. > Whether you accept this or not, the simple fact is that Cantor's > proof can be applied to any purported list of all computable Reals > and used to generate a computable Real not on the list Your "simple fact" is simply wrong. Look up the definition of "computable real" and get back to us. - Tim
From: Marshall on 18 Jun 2010 22:17 On Jun 18, 6:09 pm, Tim Little <t...(a)little-possums.net> wrote: > On 2010-06-18, Peter Webb <webbfam...(a)DIESPAMDIEoptusnet.com.au> wrote: > > > The number that is produced is clearly "computable", because we have > > computed it. > > I see you still haven't consulted a definition of "computable number". > No worries, let me know when you have. I suggest it would be more persuasive if you made whatever point you have in mind about the definition of computable number directly. Simply repeating this one-liner makes it seem like you might not have a point. Marshall
From: Sylvia Else on 18 Jun 2010 22:25
On 19/06/2010 6:50 AM, WM wrote: > On 18 Jun., 09:37, Sylvia Else<syl...(a)not.here.invalid> wrote: >> On 18/06/2010 5:31 PM, |-|ercules wrote: >> >> >> >> >> >>> "Sylvia Else"<syl...(a)not.here.invalid> wrote ... >>>> On 18/06/2010 4:52 PM, |-|ercules wrote: >>>>> "Sylvia Else"<syl...(a)not.here.invalid> wrote ... >>>>>> On 18/06/2010 3:03 PM, |-|ercules wrote: >>>>>>> "Sylvia Else"<syl...(a)not.here.invalid> wrote >>>>>>>> On 18/06/2010 10:40 AM, Transfer Principle wrote: >>>>>>>>> On Jun 17, 6:56 am, Sylvia Else<syl...(a)not.here.invalid> wrote: >>>>>>>>>> On 15/06/2010 2:13 PM, |-|ercules wrote: >>>>>>>>>>> the list of computable reals contain every digit of ALL possible >>>>>>>>>>> infinite sequences (3) >>>>>>>>>> Obviously not - the diagonal argument shows that it doesn't. >> >>>>>>>>> But Herc doesn't accept the diagonal argument. Just because >>>>>>>>> Else accepts the diagonal argument, it doesn't mean that >>>>>>>>> Herc is required to accept it. >> >>>>>>>>> Sure, Cantor's Theorem is a theorem of ZFC. But Herc said >>>>>>>>> nothing about working in ZFC. To Herc, ZFC is a "religion" >>>>>>>>> in which he doesn't believe. >> >>>>>>>> Well, if he's not working in ZFC, then he cannot make statements >>>>>>>> about >>>>>>>> ZFC, and he should state the axioms of his system. >> >>>>>>> Can you prove from axioms that is what I should do? >> >>>>>>> If you want to lodge a complaint with The Eiffel Tower that the >>>>>>> lift is >>>>>>> broken >>>>>>> do you build your own skyscraper next to the Eiffel Tower to >>>>>>> demonstrate >>>>>>> that fact? >> >>>>>> That's hardly a valid analogy. >> >>>>>> If you're attempting to show that ZFC is inconsistent, then say that >>>>>> you are working within ZFC. >> >>>>>> If you're not working withint ZFC, then you're attempting to show that >>>>>> some other set of axioms is inconsistent, which they may be, but the >>>>>> result is uninteresting, and says nothing about ZFC. >> >>>>>> Sylvia. >> >>>>> That would be like finding a fault with the plans of The Leaning Tower >>>>> Of Piza. >> >>>>> I might look at ZFC at some point, but while you're presenting Cantor's >>>>> proof >>>>> in elementary logic I'll attack that logic. >> >>>>> Instead of 'constructing' a particular anti-diagonal, your proof should >>>>> work equally >>>>> well by giving the *form* of the anti-diagonal. >> >>>>> This is what a general diagonal argument looks like. >> >>>>> For any list of reals L. >> >>>>> CONSTRUCT a real such that >>>>> An AD(n) =/= L(n,n) >> >>>>> Now to demonstrate this real is not on L, it is obvious that >>>>> An AD(n) =/= L(n,n) >> >>>>> Therefore >>>>> [ An AD(n) =/= L(n,n) -> An AD(n) =/= L(n,n) ] proves superinfinity! >> >>>>> And THAT is Cantor's proof! >> >>>>> Want to see his other proof? That no box contains the box numbers (of >>>>> boxes) that >>>>> don't contain their own box number? >>>>> That ALSO proves superinfinity! >> >>>>> Great holy grail of mathematics you have there. >> >>>>> Herc >> >>>> What are you trying to prove? >> >>> There is only one type of infinity. >> >> Infinity is a mathematical construct. Before you can even being to >> discuss it, you have to have a set of axioms. > > What was the set that Cantor used? > Nevertheless he "proved". He certainly was using some. For example, the diagonal argument falls apart if the axioms don't permit the construction of a number by choosing digits different from those on the diagonal. It isn't even clear whether Herc is tying to invalidate Cantor's proof by finding a mistake in it, or to prove the inverse, which wouldn't invalidate Cantor's proof, but would only show that the axioms on which it is based are inconsistent. Herc cannot avoid the need to specify the set of axioms. Sylvia. |